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Published online by Cambridge University Press: 06 July 2023
We study Lagrangian torus fibrations: maps on symplectic manifolds whose fibres are Lagrangian. We show they are locally the same as integrable Hamiltonian systems and then use the Arnold–Liouville theorem to construct an integral affine structure on the image of a Lagrangian torus fibration. We give an interpretation of this in terms of flux integrals and introduce the key concepts of the developing map of an integral affine structure and the affine monodromy. We also discuss the extent to which the integral affine structure on the image determines the symplectic manifold.
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